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## Main Question or Discussion Point

Hey guys how are you? I have the following question:

Let X1,X2,...,Xn be a random sample from a Pareto distribution having pdf

f(x|b)= (a*b^a)/x^(a+1) where x>=b (1)

Determine the maximum likelihood estimator for b, say b' on (0,infinity) and by considering P(b'>x) or otherwise show that b' has the Pareto distribution with pdf given by (1) but with a replaced by an.

My attempt: I found the MLE as b'=min Xi where 1<=i<=n, since our pdf is monotonically increasing w.r.t b.

After that I know how to find the asymptotic distribution of the MLE using the formula including the expected information but then we say that MLE follows a normal distribution for large n.

How do I show that the MLE follows a Pareto distribution in this case? I am so struggled, any help would be much appreciated!

P.S The hint tells us to consider P(b'>x) but how can I find P(min Xi >x) and why should it help me?

Let X1,X2,...,Xn be a random sample from a Pareto distribution having pdf

f(x|b)= (a*b^a)/x^(a+1) where x>=b (1)

Determine the maximum likelihood estimator for b, say b' on (0,infinity) and by considering P(b'>x) or otherwise show that b' has the Pareto distribution with pdf given by (1) but with a replaced by an.

My attempt: I found the MLE as b'=min Xi where 1<=i<=n, since our pdf is monotonically increasing w.r.t b.

After that I know how to find the asymptotic distribution of the MLE using the formula including the expected information but then we say that MLE follows a normal distribution for large n.

How do I show that the MLE follows a Pareto distribution in this case? I am so struggled, any help would be much appreciated!

P.S The hint tells us to consider P(b'>x) but how can I find P(min Xi >x) and why should it help me?